Question

A fixed mass of water \(m\), initially a saturated liquid, is brought to a saturated vapor condition while its pressure and temperature remain constant. (a) Derive expressions for the work and heat transfer in terms of the mass \(m\) and properties that can be obtained directly from the steam tables. (b) Demonstrate that this process is internally reversible.

Short answer

Work W = P(Vg - vf), Heat Q = mhfg. The process is internally reversible because it maintains equilibrium during the phase change.

Step-by-step solution

- Understanding the Given Conditions

Identify that the process involves water transitioning from a saturated liquid to a saturated vapor phase while keeping pressure and temperature constant.

- Determining Work Done (W)

The work done during a phase change at constant pressure can be expressed as: Work (W) = Pressure (P) × Change in Volume ( ΔV) . Since volume change ΔV = (Volume of vapor (Vg) - Volume of liquid ( vf) : W = P × (Vg - vf). Use values for Vg (specific volume of vapor) @ T and specific vf(volume of liquid @ T to obtain final work Done,W.

- Determining Heating Transfer (Q)

From steam table, the heat transfer during phase change is the latent heat of vaporization: Q = mhfg. Where hfg is the enthalpy of vaporization (from steam tables) and m is the mass of water.

- Demonstrating Internal Reversibility

For a process to be internally reversible, the system must be internally at equilibrium during the process. At equilibrium, the entropy change ( dS) is zero. In the phase change process, the system remains at equilibrium, hence the process is internally reversible.

- Final Expressions

Summarize the expressions for both work (W) and heat transfer (Q) based on the above steps: W = P (Vg - vf) and Q = mhfg.

Key concepts

saturated liquid

A saturated liquid is a state of a substance where it is about to begin vaporizing. In the context of water, this means that the liquid is at its boiling point and any addition of heat will cause it to start turning into vapor. A saturated liquid has a specific temperature and pressure at which it is stable and is also characterized by its maximum density in the liquid state. You'll find these properties listed in steam tables, which are convenient tools for finding thermodynamic properties of water.

saturated vapor

A saturated vapor occurs when a liquid has completely turned into vapor at a given pressure and temperature, without superheating. It represents a state where the vapor contains no droplets of liquid, meaning all the liquid has been converted to vapor. In steam tables, the corresponding specific volume, enthalpy, and entropy values for saturated vapor conditions are provided. It's important to note that the specific volume of the vapor is much larger than that of the liquid due to the phase change.

latent heat of vaporization

The latent heat of vaporization is the energy required to change a unit mass of a substance from a liquid to a vapor at constant temperature and pressure. For water, this value can be found in the steam tables under the term enthalpy of vaporization (h_fg). During the phase change from saturated liquid to saturated vapor, this amount of energy must be supplied without increasing either the pressure or temperature. This energy contributes to breaking the intermolecular bonds in the liquid as it transitions to the vapor state.

internal reversibility

A thermodynamic process is internally reversible if it can be reversed with an infinitesimal change without any entropy generation within the system. In practice, this means the system remains in a state of equilibrium throughout the process, ensuring there is no loss of available energy. In the example of water transitioning from a saturated liquid to a saturated vapor, the process is internally reversible because it occurs at constant pressure and temperature, maintaining equilibrium conditions.

steam tables

Steam tables are comprehensive charts that provide the thermodynamic properties of water in both liquid and vapor phases across a range of temperatures and pressures. They are essential tools in thermodynamics for finding properties such as specific volume, internal energy, enthalpy, and entropy. For instance, during the phase transition of water from saturated liquid to saturated vapor, steam tables provide the values for Vg (specific volume of vapor), vf (specific volume of liquid), and h_fg (latent heat of vaporization). These values are crucial for calculating work done and heat transfer in thermodynamic processes.